In this paper, we consider the problem of robust M-estimation of parameters of nonlinear signal processing models. We investigate the conditions under which estimators are strongly consistent for convex and non-convex penalty functions and a wide class of noise scenarios, contaminating the actual transmitted signal. It is shown that the M-estimators of a general nonlinear signal model are asymptotically consistent with probability one under different sets of sufficient conditions on loss function and noise distribution. Simulations are performed for nonlinear superimposed sinusoidal model to observe the small sample performance of the M-estimators for various heavy tailed error distributions, outlier contamination levels and sample sizes.
In both statistical non-inferiority (NI) and superiority (S) tests, the critical region must be a Barnard convex set for two main reasons. One, being computational in nature, based on the fact that calculating test sizes is a computationally intensive problem due to the presence of a nuisance parameter. However, this calculation is considerably reduced when the critical region is a Barnard convex set. The other reason is that in order for the NI/S statistical tests to make sense, its critical regions must be Barnard convex sets. While it is indeed possible for NI/S tests’ critical regions to not be Barnard convex sets, for the reasons stated above, it is desirable that they are. Therefore, it is important to generate, from a given NI/S test, a test which guarantees that the critical regions are Barnard convex sets. We propose a method by which, from a given NI/S test, we construct another NI/S test, ensuring that the critical regions corresponding to the modified test are Barnard convex sets, we illustrate this through examples. This work is theoretical because the type of developments refers to the general framework of NI/S testing for two independent binomial proportions and it is applied because statistical tests that do not ensure that their critical regions are Barnard convex sets may appear in practice, particularly in the clinical trials area.
The odds ratio is the predominant measure of association in 2×2 contingency tables, which, for inferential purposes, is usually considered on the log-scale. Under an information theoretic set-up, it is connected to the Kullback–Leibler divergence. Considering a generalized family of divergences, the ϕ divergence, alternative association measures are derived for 2×2 contingency tables. Their properties are studied and asymptotic inference is developed. For some members of this family, the estimated association measures remain finite in the presence of a sampling zero while for a subset of these members the estimators of these measures have finite variance as well. Special attention is given to the power divergence, which is a parametric family. The role of its parameter λ, in terms of the asymptotic confidence intervals' coverage probability and average relative length, is further discussed. In special probability table structures, for which the performance of the asymptotic confidence intervals for the classical log odds ratio is poor, the measure corresponding to λ=1/3 is suggested as an alternative.
A new stochastic order called dynamic cumulative residual quantile entropy (DCRQE) order is established. Some characterizations of the new order are investigated. Closure and reversed closure properties of the DCRQE order are obtained. Applications of the DCRQE ordering in characterizing the proportional hazard rate model and the k-record values model are considered.
We propose a new variable selection and estimation technique for high dimensional single index models with unknown monotone smooth link function. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection. In this article, we propose a new penalized forward selection technique which can reduce high dimensional optimization problems to several one dimensional optimization problems by choosing the best predictor and then iterating the selection steps until convergence. The advantage of optimizing in one dimension is that the location of optimum solution can be obtained with an intelligent search by exploiting smoothness of the criterion function. Moreover, these one dimensional optimization problems can be solved in parallel to reduce computing time nearly to the level of the one-predictor problem. Numerical comparison with the LASSO and the shrinkage sliced inverse regression shows very promising performance of our proposed method.
This research is motivated from the analysis of a real gene expression data that aims to identify a subset of "interesting" or "significant" genes for further studies. When we blindly applied the standard false discovery rate (FDR) methods, our biology collaborators were suspicious or confused, as the selected list of significant genes was highly unbalanced: there were ten times more under-expressed genes than the over-expressed genes. Their concerns led us to realize that the observed two-sample t-statistics were highly skewed and asymmetric, and thus the standard FDR methods might be inappropriate. To tackle this case, we propose a symmetric directional FDR control method that categorizes the genes into "over-expressed" and "under-expressed" genes, pairs "over-expressed" and "under-expressed" genes, defines the p-values for gene pairs via column permutations, and then applies the standard FDR method to select "significant" gene pairs instead of "significant" individual genes. We compare our proposed symmetric directional FDR method with the standard FDR method by applying them to simulated data and several well-known real data sets.
Shrinkage estimation is used to develop a semiparametric density estimator as a linear combination of a fully known parametric density function and a nonparametric density estimator. We determine the asymptotic properties of the shrinkage coefficient and of the semiparametric estimator’s integrated squared error. Moreover, we show that the proposed estimation methodology delivers density estimators that are more accurate than nonparametric estimators and that do not require the use of optimal smoothing parameters.
A problem for estimating the number of trials n in the binomial distribution B(n,p), is revisited by considering the large sample model N(μ,cμ) and the associated maximum likelihood estimator (MLE) and some sequential procedures. Asymptotic properties of the MLE of n via the normal model N(μ,cμ) are briefly described. Beyond the asymptotic properties, our main focus is on the sequential estimation of n. Let X1,X2,…,Xm,… be iid N(μ,cμ)(c>0) random variables with an unknown mean μ=1,2,… and variance cμ, where c is known. The sequential estimation of μ is explored by a method initiated by Robbins (1970) and further pursued by Khan (1973). Various properties of the procedure including the error probability and the expected sample size are determined. An asymptotic optimality of the procedure is given. Sequential interval estimation and point estimation are also briefly discussed.
The testing of equality of several Pearson correlations can be found in a number of scientific fields. We surmise in many such cases that the alternatives of interest in practice are, in deed, order restricted, and therefore the researcher is best served by use of testing procedures developed for those specific alternatives. In this note we introduce a collection of tests for use in testing equality of k correlation coefficients against order alternatives, with an emphasis on simple order. Specifically, we propose likelihood ratio tests and contrast tests based on the well known Fisher Z transformation as well as tests which make use of generalized variable methodologies. The proposed procedures are empirically compared with regard to type I and II error rates via Monte Carlo simulations studies, and the use of the approaches is illustrated using an example. These tests are found to be vastly superior to tests for the general alternative, and the contrast tests based on the Fisher Z transformation are recommended for practice based on the observed test properties and simplicity. (C) 2016 Elsevier B.V. All rights reserved.
Kernel-based association test (KAT) is a widely used tool in genetics association analysis. The performance of such a test depends on the choice of kernel. In this paper, we study the statistical power of a KAT using a Gaussian kernel. We explicitly develop a notion of analytical power function in this family of tests. We propose a novel approach to select the kernel so as to maximize the analytical power function of the test at a given test level (an upper bound on the probability of making a type I error). We assess some theoretical properties of our optimal estimator, and compare its performance with some similar existing alternatives using simulation studies. Neuroimaging data from an Alzheimer’s disease study is also used to illustrate the proposed kernel selection methodology.
In this paper, we discuss continuous time periodic debugging of a software, in which faults are corrected, instead of at the instants of their detection, at some prespecified times. We describe maximum-likelihood estimation for the model parameters assuming exponential distribution for the time between successive occurrence of a fault, which can be used to estimate the reliability of the software. We also derive the asymptotic properties of the estimated model parameters, specially the number of faults initially present in the software. We investigate the finite sample properties of the estimators, specially that of the number of faults, through simulation. Finally, we analyze a dataset obtained from the testing of a flight control software.
We consider two types of graphs based on a family of proximity catch digraphs (PCDs) and study their edge density. In particular, the PCDs we use are a parameterized digraph family called proportional-edge (PE) PCDs and the two associated graph types are the “underlying graphs” and the newly introduced “reflexivity graphs” based on the PE-PCDs. These graphs are extensions of random geometric graphs where distance is replaced with a dissimilarity measure and the threshold is not fixed but depends on the location of the points. PCDs and the associated graphs are constructed based on data points from two classes, say X and Y, where one class (say class X) forms the vertices of the PCD and the Delaunay tessellation of the other class (i.e., class Y) yields the (Delaunay) cells which serve as the support of class X points. We demonstrate that edge density of these graphs is a U-statistic, hence obtain the asymptotic normality of it for data from any distribution that satisfies mild regulatory conditions. The rate of convergence to asymptotic normality is sharper for the edge density of the reflexivity and underlying graphs compared to the arc density of the PE-PCDs. For uniform data in Euclidean plane where Delaunay cells are triangles, we demonstrate that the distribution of the edge density is geometry invariant (i.e., independent of the shape of the triangular support). We compute the explicit forms of the asymptotic normal distribution for uniform data in one Delaunay triangle in the Euclidean plane utilizing this geometry invariance property. We also provide various versions of edge density in the multiple triangle case. The approach presented here can also be extended for application to data in higher dimensions.
Acute hypotensive episodes (AHEs) are serious clinical events in intensive care units (ICUs), and require immediate treatment to prevent patient injury. Reducing the risks associated with an AHE requires effective and efficient mining of data generated from multiple physiological time series. We propose HeartCast, a model that extracts essential features from such data to effectively predict AHE. HeartCast combines a non-linear support vector machine with best-feature extraction via analysis of the baseline threshold, quartile parameters, and window size of the physiological signals. Our approach has the following benefits: (a) it extracts the most relevant features; (b) it provides the best results for identification of an AHE event; (c) it is fast and scales with linear complexity over the length of the window; and (d) it can manage missing values and noise/outliers by using a best-feature extraction method. We performed experiments on data continuously captured from physiological time series of ICU patients (roughly 3 GB of processed data). HeartCast was found to outperform other state-of-the-art methods found in the literature with a 13.7% improvement in classification accuracy.
In this paper, we consider generalized inverted exponential distribution which is capable of modeling various shapes of failure rates and aging criteria. The purpose of this paper is two fold. Based on progressive type-II censored data, first we consider the problem of estimation of parameters under classical and Bayesian approaches. In this regard, we obtain maximum likelihood estimates, and Bayes estimates under squared error loss function. We also compute 95% asymptotic confidence interval and highest posterior density interval estimates under the respective approaches. Second, we consider the problem of prediction of future observations using maximum likelihood predictor, best unbiased predictor, conditional median predictor and Bayes predictor. The associated predictive interval estimates for the censored observations are computed as well. Finally, we analyze two real data sets and conduct a Monte Carlo simulation study to compare the performance of the various proposed estimators and predictors.
A flexible class of skew-slash distributions which is a location-scale mixture of skew-elliptically distributed random variable with power of a beta random variable is presented. This family of distributions, which is a generalization of location-scale mixture of normal and beta distributions, contain some existing and important distributions and is appropriate for modeling data with skewness and heavy tail structure. Some distributional properties and the moments of this new family of distributions are obtained. In the special case of location-scale mixture of skew-normal distribution, we estimate the parameters via an EM-type algorithm and a simulation study and an application to real data are provided for illustration. Finally we extend some results to multivariate case.
Recently, the selection consistency of penalized least square estimators has received a great deal of attention. For the penalized likelihood estimation with certain non-convex penalties, search space can be constructed within which there exists a unique local minimizer that exhibits selection consistency in high-dimensional generalized linear models under certain conditions. In particular, we prove that the SCAD penalty of Fan and Li (2001) and a new modified version of the unbounded penalty of Lee and Oh (2014) can be employed to achieve such a property. These results hold even for the non-sparse cases where the number of relevant covariates increases with the sample size. Simulation studies are provided to compare the performance of SCAD penalty and the newly proposed penalty.
The analysis of longitudinal data or repeated measurements is an important and growing area of Statistics. In this context, data come in different formats but typically, they have a hierarchical or multi-level structure including group and subject components, and the main purpose of the analysis is usually to estimate these components from the data. A standard way to perform this estimation is via mixed models. In this paper, we show that the estimated group effects from standard smooth mixed models can deviate systematically from the underlying group mean, leading to wrong conclusions about the data. We then present two ways to avoid such systematic deviations and misinterpretations when fitting flexible mixed models to multi-level data. The first method is a marginal procedure, and the second method is based on the conditional distribution of the subject effects derived from appropriate constraints. Both methods are robust against mis-specification of the covariance structure in the sense that they allow one to resolve the lack of centring found in standard smooth mixed models.
In this study a varying-coefficient partially nonlinear model with measurement errors in the nonparametric part is proposed. Based on the corrected profile least-squared estimation methodology, we define the estimates of the unknowns of the current models, and check whether the coefficient functions are a constant or not by using the popular generalized likelihood ratio (GLR) test method. Further, the corresponding asymptotic distribution is established and a bootstrap procedure is also employed to implement the proposed methodology. Simulated and real examples are given to illustrate our proposed methodology. (C) 2016 Elsevier B.V. All rights reserved.
Based on stochastically independent samples with underlying density functions from the same multiparameter exponential family, a weighted version of Matusita’s affinity is applied as test statistic in a homogeneity test of identical densities as well as in a discrimination problem. Asymptotic distributions of the test statistics are stated, and the impact of weights on the deviation of actual and required type I error for finite sample sizes is examined in a simulation study.
We derive two new confidence ellipsoids (CEs) and four CE variations for covariate coefficient vectors with nuisance parameters under the seemingly unrelated regression (SUR) model. Unlike most CE approaches for SUR models studied so far, we assume unequal regression coefficients for our two regression models. The two new basic CEs are a CE based on a Wald statistic with nuisance parameters and a CE based on the asymptotic normality of the SUR two-stage unbiased estimator of the primary regression coefficients. We compare the coverage and volume characteristics of the six SUR-based CEs via a Monte Carlo simulation. For the configurations in our simulation, we determine that, except for small sample sizes, a CE based on a two-stage statistic with a Bartlett corrected (1−α) percentile is generally preferred because it has essentially nominal coverage and relatively small volume. For small sample sizes, the parametric bootstrap CE based on the two-stage estimator attains close-to-nominal coverage and is superior to the competing CEs in terms of volume. Finally, we apply three SUR Wald-type CEs with favorable coverage properties and relatively small volumes to a real data set to demonstrate the gain in precision over the ordinary-least-squares-based CE.
